A144304 Square array A(n,m), n>=0, m>=0, read by antidiagonals: A(n,m) = n-th number of the m-th iteration of the hyperbinomial transform on sequence A001858.

1, 1, 1, 1, 2, 2, 1, 3, 7, 7, 1, 4, 14, 38, 38, 1, 5, 23, 93, 291, 291, 1, 6, 34, 178, 822, 2932, 2932, 1, 7, 47, 299, 1763, 9193, 36961, 36961, 1, 8, 62, 462, 3270, 21504, 125292, 561948, 561948, 1, 9, 79, 673, 5523, 43135, 313585, 2022555, 10026505, 10026505, 1

Offset

0,5

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

N. J. A. Sloane, Transforms

Example

Square array begins:
   1,   1,   1,    1,    1, ...
   1,   2,   3,    4,    5, ...
   2,   7,  14,   23,   34, ...
   7,  38,  93,  178,  299, ...
  38, 291, 822, 1763, 3270, ...

Maple

hymtr:= proc(p) proc(n, m) `if`(m=0, p(n), m*add(p(k) *binomial(n, k) *(n-k+m)^(n-k-1), k=0..n)) end end: f:= proc(n) option remember; add(add(binomial(m, j) *binomial(n-1, n-m-j) *n^(n-m-j) *(m+j)!/ (-2)^j/ m!, j=0..m), m=0..n) end: A:= hymtr(f): seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

hymtr[p_] := Function[{n, m}, If[m == 0, p[n], m*Sum[p[k]*Binomial[n, k]*(n-k+m)^(n-k-1), {k, 0, n}]]]; f[0] = 1; f[n_] := f[n] = Sum[Sum[Binomial[m, j]*Binomial[n-1, n-m-j]*n^(n-m-j)*(m+j)!/(-2)^j/m!, {j, 0, m}], {m, 0, n}]; A[0, _] = 1; A[1, k_] := k+1; A[n_, m_] := hymtr[f][n, m]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

Crossrefs

Columns m=0-3 give: A001858, A001858(n+1), A089462, A089465.

Rows n=0-2 give: A000012, A000027, A008865(m+2).

Main diagonal gives A252727.

Sequence in context: A007441 A289192 A111933 * A122941 A297622 A059584

Adjacent sequences: A144301 A144302 A144303 * A144305 A144306 A144307

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 17 2008

Status

approved